Preframe presentations present

  • Peter Johnstone
  • Steven Vickers
Part I
Part of the Lecture Notes in Mathematics book series (LNM, volume 1488)


Preframes (directed complete posets with finite meets that distribute over the directed joins) are the algebras for an infinitary essentially algebraic theory, and can be presented by generators and relations. This result is combined with a general argument concerning categories of commutative monoids to give a very short proof of the localic Tychonoff Theorem.

It is also shown how frames can be presented as preframes, a result analogous to Johnstone's construction of frames from sites, and an application is given.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

6. Bibliography

  1. [90]
    S. ABRAMSKY and S.J. VICKERS Quantales, Observational Logic and Process Semantics, Department of Computing Report DOC 90/1, Imperial College of Science, Technology and Medicine, London, 1990.MATHGoogle Scholar
  2. [88]
    B. BANASCHEWSKI “Another look at the Localic Tychonoff Theorem”, pp. 647–656 in Commentationes Mathematicae Universitatis Carolinae 29(4), Charles University, Prague, 1988.Google Scholar
  3. [70]
    N. BOURBAKI Éléments de Mathématique: Algèbre, Hermann, Paris, 1970.MATHGoogle Scholar
  4. [90]
    T. COQUAND “An intuitionistic proof of Tychonoff's theorem”, preprint (1990).Google Scholar
  5. [79]
    M. COSTE “Localization, spectra and sheaf representation”, pp. 212–238 in M.P.Fourman, C.J.Murvey and D.S.Scott (eds) Applications of Sheaves, Springer Lecture Notes in Mathematics 753, 1979.Google Scholar
  6. [76]
    C.H. DOWKER and D. STRAUSS “Sums in the category of frames”, pp. 17–32 in Houston J. Math. 3 (1976).MathSciNetMATHGoogle Scholar
  7. [57]
    C. EHRESMANN “Gattungen von lokalen Strukturen”, pp. 59–77 in Jber. Deutsch. Math.-Verein. 60 (1957).MathSciNetMATHGoogle Scholar
  8. [72]
    P. FREYD “Aspects of topoi”, pp. 1–76 in Bull. Austral. Math. Soc. 7 (1972).MathSciNetCrossRefMATHGoogle Scholar
  9. [90]
    P. FREYD and A. SCEDROV Categories, Allegories, North-Holland 1990.Google Scholar
  10. [71]
    P. GABRIEL and F. ULMER “Lokal präsentierbare Kategorien”, Springer Lecture Notes in Mathematics 221, 1971.Google Scholar
  11. [80]
    G. GIERZ, K.H. HOFMANN, K. KEIMEL, J.D. LAWSON, M. MISLOVE and D.S. SCOTT A Compendium of Continuous Lattices, Springer-Verlag, Berlin, 1980.CrossRefMATHGoogle Scholar
  12. [81]
    KARL H. HOFMANN and MICHAEL W. MISLOVE “Local compactness and continuous lattices”, pp. 209–248 in B. Banaschewski and R.-E. Hoffmann (eds) Continuous Lattices: Proceedings, Bremen 1979, Lecture Notes in Mathematics 871, Springer-Verlag, Berlin, 1981.CrossRefGoogle Scholar
  13. [72]
    J.R. ISBELL “General functorial semantics I”, pp. 535–596 in Amer. J. Math 94 (1972).MathSciNetCrossRefMATHGoogle Scholar
  14. [81]
    P.T. JOHNSTONE “Tychonoff's theorem without the axiom of choice”, pp. 21–35 in Fund. Math. 113 (1981).MathSciNetMATHGoogle Scholar
  15. [82]
    Stone Spaces, Cambridge University Press, Cambridge, 1982.MATHGoogle Scholar
  16. [85]
    “Vietoris locales and localic semilattices”, pp. 155–180 in R.-E. Hoffmann and K.H. Hofmann (eds) Continuous Lattices and their Applications, Marcel Dekker 1985.Google Scholar
  17. [88]
    P.T. JOHNSTONE and S.H. SUN “Weak products and Hausdorff locales”, pp. 173–193 in F. Borceux (ed.) Categorical Algebra and its Applications, Springer Lecture Notes in Mathematics 1348, 1988.Google Scholar
  18. [84]
    ANDRÉ JOYAL and MYLES TIERNEY An Extension of the Galois Theory of Grothendieck, Memoirs of the American Mathematical Society 309, 1984.Google Scholar
  19. [85]
    I. KŘÍŽ “A constructive proof of the Tychonoff theorem for locales”, pp. 619–630 in Commentationes Mathematicae Universitatis Carolinae 26, Charles University, Prague, 1985.Google Scholar
  20. [66]
    F.E.J. LINTON “Some aspects of equational categories”, pp. 84–94 in S.Eilenberg et al. (eds) Proceedings of the Conference on Categorical Algebra, Springer-Verlag 1966.Google Scholar
  21. [69]
    “Coequalizers in categories of algebras”, pp. 75–90 in Seminar on Triples and Categorical Homology Theory, Springer Lecture Notes in Mathematics 80, 1969.Google Scholar
  22. [82]
    J.L. MacDONALD and A. STONE “The tower and regular decomposition”, pp. 197–213 in Cahiers Top. Géom. Diff. 23 (1982).MathSciNetMATHGoogle Scholar
  23. [86]
    C. McLARTY “Left exact logic”, pp. 63–66 in J. Pure Appl. Alg. 41 (1986).MathSciNetCrossRefMATHGoogle Scholar
  24. [51]
    ERNEST MICHAEL “Topologies on spaces of subsets”, pp. 152–182 in Trans. Amer. Math. Soc. 71 (1951).MathSciNetCrossRefMATHGoogle Scholar
  25. [64]
    S. PAPERT “An abstract theory of topological spaces”, pp. 197–203 in Proc. Camb. Phil. Soc. 60 (1964).MathSciNetCrossRefMATHGoogle Scholar
  26. [59]
    J. SLOMINSKI “The theory of abstract algebras with infinitary operations”, Rozprawy Mat.18 (1959).Google Scholar
  27. [78]
    M. SMYTH “Power domains”, in JCSS 16, 1978.Google Scholar
  28. [83]
    “Powerdomains and predicate transformers: a topological view”, pp. 662–675 in J. Diaz (ed.) Automata, Languages and Programming, Lecture Notes in Computer Science 154, Springer-Verlag, Berlin, 1983.CrossRefGoogle Scholar
  29. [90]
    J.J.C VERMEULEN “Tychonoff's theorem in a topos”, preprint (1990.Google Scholar
  30. [89]
    STEVEN VICKERS Topology via Logic, Cambridge University Press, Cambridge, 1989.MATHGoogle Scholar
  31. [79]
    D. WIGNER “Two notes on frames”, pp. 257–268 in J.Austral.Math.Soc. A28 (1979).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Peter Johnstone
    • 1
  • Steven Vickers
    • 2
  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridge
  2. 2.Department of ComputingImperial CollegeLondon

Personalised recommendations